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Hodgkin–Huxley-Type Models of Cardiac Muscle Cells

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Computational Electrophysiology

Part of the book series: A First Course in “In Silico Medicine” ((FCISM,volume 2))

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Abstract

Following the HH formalism introduced in Chap. 2, various kinds of HH-type models of neurons and other excitable cells are proposed (Canavier et al. 1991; Chay and Keizer 1983; Cronin 1987; Gerber and Jakobsson 1993; Hayashi and Ishizuka 1992; Keener and Sneyd 1998; Noble 1995; Rinzel 1990; Traub et al. 1991), and are analyzed (Alexander and Cai 1991; Av-Ron 1994; Bertram 1994; Bertram et al. 1995; Butera 1998; Canavier et al. 1993; Chay and Rinzel 1985; Doi and Kumagai 2005; Guckenheimer et al. 1993; Maeda et al. 1998; Rush and Rinzel 1994; Schweighofer et al. 1999; Terman 1991; Tsumoto et al. 2003, 2006; Yoshinaga et al. 1999). The HH-type equations include many variables depending on the number of different ionic currents and their gating variables considered in the equations, whereas the original HH equations possess only four variables (a membrane voltage, activation and inactivation variables of Na+ current and an activation variable of K+ current). Among the diverse family of HH-type equations, this chapter explores the dynamics and the bifurcation structure of the HH-type equations of heart muscle cells (cardiac myocytes).

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References

  • Alexander JC, Cai DY (1991) On the dynamics of bursting systems. J Math Biol 29:405–423

    Article  MathSciNet  MATH  Google Scholar 

  • Av-Ron E (1994) The role of a transient potassium current in a bursting neuron model. J Math Biol 33:71–87

    Article  MATH  Google Scholar 

  • Beeler GW, Reuter H (1977) Reconstruction of the action potential of ventricular myocardial fibres. J Physiol (London) 268:177–210

    Google Scholar 

  • Bertram R (1994) Reduced-system analysis of the effects of serotonin on a molluscan burster neuron. Biol Cybern 70:359–368

    Article  MATH  Google Scholar 

  • Bertram R, Butte MJ, Kiemel T, Sherman A (1995) Topological and phenomenological classification of bursting oscillations. Bull Math Biol 57:413–439

    MATH  Google Scholar 

  • Butera RJ Jr (1998) Multirhythmic bursting. Chaos 8:274–284

    Article  MathSciNet  MATH  Google Scholar 

  • Canavier CC, Clark JW, Byrne JH (1991) Simulation of the bursting activity of neuron R15 in Aplisia: role of ionic currents, calcium balance, and modulatory transmitters. J Neurophysiol 66:2107–2124

    Google Scholar 

  • Canavier CC, Baxter DA, Clark JW, Byrne JH (1993) Nonlinear dynamics in a model neuron provide a novel mechanism for transient synaptic inputs to produce long-term alterations of postsynaptic activity. J Neurophysiol 69:2252–2257

    Google Scholar 

  • Chay TR, Keizer J (1983) Minimal model for membrane oscillations in the pancreatic β-cell. Biophys J 42:181–190

    Article  Google Scholar 

  • Chay TR, Rinzel J (1985) Bursting, beating, and chaos in an excitable membrane model. Biophys J 47:357–366

    Article  Google Scholar 

  • Cronin J (1987) Mathematical aspects of Hodgkin–Huxley neural theory. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • DiFrancesco D, Noble D (1985) A model of cardiac electrical activity incorporating ionic pumps and concentration changes. Philos Trans R Soc Lond [Biol] 307:353–398

    Article  Google Scholar 

  • Doi S, Kumagai S (2005) Generation of very slow neuronal rhythms and chaos near the Hopf bifurcation in single neuron models. J Comp Neurosci 19:325–356

    Article  MathSciNet  Google Scholar 

  • Gerber B, Jakobsson E (1993) Functional significance of the A-current. Biol Cybern 70:109–114

    Article  Google Scholar 

  • Guckenheimer J, Gueron S, Harris-Warrick RM (1993) Mapping the dynamics of a bursting neuron. Phil Trans R Soc Lond B 341:345–359

    Article  Google Scholar 

  • Hayashi H, Ishizuka S (1992) Chaotic nature of bursting discharges in the Onchidium pacemaker neuron. J Theor Biol 156:269–291

    Article  Google Scholar 

  • Hilgemann DW, Noble D (1987) Excitation-contraction coupling and extracellular calcium transients in rabbit atrium: reconstruction of the basic cellular mechanisms. Proc R Soc Lond B Biol Sci 230:163–205

    Article  Google Scholar 

  • Keener JP, Sneyd J (1998) Mathematical physiology. Springer, Berlin

    MATH  Google Scholar 

  • Luo CH, Rudy Y (1991) A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. Circ Res 68:1501–1526

    Article  Google Scholar 

  • Luo CH, Rudy Y (1994) A dynamic model of the ventricular cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ Res 74:1071–1096

    Google Scholar 

  • Maeda Y, Pakdaman K, Nomura T, Doi S, Sato S (1998) Reduction of a model for an Onchidium pacemaker neuron. Biol Cybern 78:265–276

    Article  MATH  Google Scholar 

  • McAllister RE, Noble D, Tsien RW (1975) Reconstruction of the electrical activity of cardiac Purkinje fibres. J Physiol (London) 251:1–59

    Google Scholar 

  • Noble D (1962) Modification of Hodgkin–Huxley equations applicable to purkinje fibre action and pace-maker potentials. J Physiol (London) 160:317–352

    Google Scholar 

  • Noble D (1975) The initiation of the heartbeat. Oxford University Press, Oxford

    Google Scholar 

  • Noble D (1995) The development of mathematical models of the heart. Chaos Solitons Fractals 5:321–333

    Article  MATH  Google Scholar 

  • Noble D, Noble SJ (1984) A model of sino-atrial node electrical activity based on a modification of the DiFrancesco–Noble (1984) equations. Proc R Soc Lond B Biol Sci 222:295–304

    Article  Google Scholar 

  • Priebe L, Beuckelmann DJ (1998) Simulation study of cellular electric properties in heart gailure. Circ Res 82:1206–1223

    Article  Google Scholar 

  • Ramirez RJ, Nattel S (2000) Courtemanche M: Mathematical analysis of canine atrial action potentials: rate, regional factors, and electrical remodeling. Am J Physiol Heart Circ Physiol 279:H1767–H1785

    Google Scholar 

  • Rinzel J (1990) Discussion: electrical excitability of cells, theory and experiment: review of the Hodgkin–Huxley foundation and update. Bull Math Biol 52:5–23

    Article  Google Scholar 

  • Rush ME, Rinzel J (1994) Analysis of bursting in a thalamic neuron model. Biol Cybern 71:281–291

    Article  MATH  Google Scholar 

  • Sarai N, Matsuoka S, Kuratomi S, Ono K, Noma A (2003) Role of individual ionic current systems in the SA node hypothesized by a model study. Jpn J Physiol 53:125–134

    Article  Google Scholar 

  • Schweighofer N, Doya K, Kawato M (1999) Electrophysiological properties of inferior olive neurons: a compartmental model. J Neurophysiol 82:804–817

    Google Scholar 

  • ten Tusscher KHW, Panfilov AV (2006a) Alternans and spiral breakup in a human ventricular tissue model. Am J Physiol Heart Circ Physiol 291:H1088–H1100

    Article  Google Scholar 

  • ten Tusscher KHW, Noble D, Noble PJ, Panfilov AV (2004) A model for human ventricular tissue. Am J Physiol Heart Circ Physiol 286:H1573–H1589

    Article  Google Scholar 

  • Terman D (1991) Chaotic spikes arising from a model of bursting in excitable membranes. SIAM J Appl Math 51:1418–1450

    Article  MathSciNet  MATH  Google Scholar 

  • Traub RD, Wong RKS, Miles R, Michelson H (1991) A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances. J Neurophysiol 66:635–650

    Google Scholar 

  • Tsumoto K, Yoshinaga T, Aihara K, Kawakami H (2003) Bifurcations in synaptically coupled Hodgkin–Huxley neurons with a periodic input. Int J Bifurcat Chaos 13:653–666

    Article  MathSciNet  MATH  Google Scholar 

  • Yamaguchi R, Doi S, Kumagai S (2007) Bifurcation analysis of a detailed cardiac cell model and drug sensitivity of ionic channels. In: Proc. 15th IEEE international workshop on Nonlinear Dynamics of Electronic Systems 2007, pp 205–208

    Google Scholar 

  • Yanagihara K, Noma A, Irisawa H (1980) Reconstruction of sinoatrial node pacemaker potential based on the voltage clamp experiments. Jpn J Physiol 30:841–857

    Article  Google Scholar 

  • Yoshinaga T, Sano Y, Kawakami H (1999) A method to calculate bifurcations in synaptically coupled Hodgkin–Huxley equations. Int J Bifurcat Chaos 9:1451–1458

    Article  MATH  Google Scholar 

  • Zhang H, Holden AV, Kodama I, Honjo H, Lei M, Varghese T, Boyett MR (2000) Mathematical models of action potential in the periphery and center of the rabbit sinoatrial node. Am J Physiol 279:H397–H421

    Google Scholar 

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Authors and Affiliations

  1. Graduate School of Engineering, Kyoto University, Kyoto-Daigaku Katsura, Nishikyo-ku, Kyoto, 615-8510, Japan

    Shinji Doi

  2. Faculty of Human Science, Kyoto Koka Women’s University, 38 Kadono-cho, Nishikyogoku, Ukyo-ku, Kyoto, 615-0882, Japan

    Junko Inoue

  3. Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka, 565-0871, Japan

    Zhenxing Pan

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  1. Shinji Doi
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  2. Junko Inoue
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  3. Zhenxing Pan
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Doi, S., Inoue, J., Pan, Z. (2010). Hodgkin–Huxley-Type Models of Cardiac Muscle Cells. In: Computational Electrophysiology. A First Course in “In Silico Medicine”, vol 2. Springer, Tokyo. https://doi.org/10.1007/978-4-431-53862-2_5

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  • DOI: https://doi.org/10.1007/978-4-431-53862-2_5

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-53861-5

  • Online ISBN: 978-4-431-53862-2

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